Numerical methods for unsteady incompressible flows using primitive variables and non-staggered grids

New implicit finite-difference schemes for solving the time-dependent incompressible Navier–Stokes equations using primitive variables and non-staggered grids are presented in this paper. A priori estimate for the discrete solution of the methods is obtained. Employing the operator approach, some requirements on difference operators of a scheme are formulated in order to derive the scheme which is essentially consistent with the initial differential equations. Operators of the scheme heritage the fundamental properties of corresponding differential operators and this allows a priori estimates for the discrete solution to be obtained. To derive the consistent scheme, special approximations for convective terms, div and grad operators are employed. Used spatial approximations are of the second order within computational domain and of the first order at the boundary. The checkerboard solution oscillations are illustrated for the case that the discrete continuity equation is solved in unmodified form. To obtain a smooth discrete solution, a scheme with additional regularizing terms is derived. The terms are proportional to $O(\tau h^2)$. It is shown that the derived scheme has a very weak restriction on the time-step size. A lid-driven cavity flow has been predicted to examine stability and accuracy of the scheme for Reynolds number up to 3200 on the sequence of grids with $21\times21$, $41\times41$, $81\times81$ and $161\times161$ grid points.